High-temperature superconductivity remains arguably the greatest enigma of condensed matter physics. The discovery of iron-based high-temperature superconductors [1, 2] has renewed the importance of understanding superconductivity in materials susceptible to magnetic order and fluctuations. Intriguingly, they show magnetic fluctuations reminiscent of superconducting (SC) cuprates [3], including a 'resonance' and an 'hourglass'-shaped dispersion [4], which provides an opportunity to gain new insights into the coupling between spin fluctuations and superconductivity. In this paper, we report inelastic neutron scattering data on Fe1+yTe0.7Se0.3 using excess iron concentration to tune between an SC (y = 0.02) and a non-SC (y = 0.05) ground state. We find incommensurate spectra in both the samples but discover that in the one that becomes SC, a constriction toward a commensurate hourglass-shape develops well above T-c. Conversely, a spin gap and a concomitant spectral weight shift happen below T-c. Our results imply that the hourglass-shaped dispersion is most likely a prerequisite for superconductivity, whereas the spin gap and shift of spectral weight are the consequences of superconductivity. We explain this observation by pointing out that an inward dispersion toward the commensurate wave vector is needed for the opening of a spin gap to lower the magnetic exchange energy and hence provide the necessary condensation energy for the SC state to emerge.